Temperament: A
Beginner's Guide

By Stephen Bicknell -
oneskull@dircon.co.uk

April 25, 1997 Revised May 14, 1997

Temperament is one of those subjects
that attracts a) buffs and b) incomprehensible jargon.
The usual explanations start with an analysis of the
dreaded comma  whether Pythagorean or Syntonic 
and most of us never really get past the problem of
trying to work out what on earth such a thing might be.

I hope, kind readers, that you will
allow me to come to your rescue, and attempt to explain
in simple terms what the temperament enthusiasts are on
about.

There is a problem to be solved in
tuning any musical instrument: the notes cannot be made
to fit into the octave, and some have to be de-tuned to
make sense. If you tune a circle of pure fifths 

c - g - d - a - e - b -
f# - c# - g# - d# - a# - f - c

 the c you end up at is not
exactly in tune with the one you started with  a
mathematical anomaly known as a comma.

In equal temperament all the notes in
the scale are shifted by the same amount in order
to resolve the problem.

In all other temperaments the notes in
the scale are shifted by differing amounts, giving
each temperament a certain character. These can be
arranged in more or less chronological order:

The earliest is Pythagorean temperament, which seems to have been in use up
to the end of the 16th century. Almost all the fourths
and fifths are dead in tune, and the entire comma is
'dumped' on one interval (according to Arnaut de Zwolle
between F and Bb), which is therefore unusable. This
temperament is easy to explain and to tune, but it leaves
a lot of the notes of the scale in quite odd positions.
It is quite satisfactory for music written in the old
'modes' that preceded the major and minor scales,
provided there is no modulation whatever.

By the early 17th century meantone temperament was the norm. In this temperament
the major thirds are perfectly in tune and the fourths
and fifths slightly compromised  except for one
hideously catastrophic fifth, usually between G# and Eb,
the famous 'wolf.' However, this is now a 'regular'
temperament, for in keys with less than four accidentals
the notes of the major scale are in the same relative
positions, the thirds all pure. This, for the first time,
allows the composer freedom to include harmonic
modulation in one direction or another, and to choose a
key that mirrors his thoughts. However, during the course
of modulatory passages there will be an audible 'shift'
of tonality, rather like changing gear. The appearance of
a black note that is technically 'unavailable' in music
of the 17th century (they are A flat, A sharp, D flat, D
sharp and G flat) is a sure indication that a sudden
clash was intended  rather like the deliberate use
of false relations. The more extreme accidentals (C flat
and onwards) barely ever appear. The occasional
appearance in the mean-tone era of keys like F minor
suggests the dawning of an awareness of the possibilities
of key-colour: with four flats it has a very strange
minor third (G sharp, not A flat) and if the G flat is
called for there is further trouble in store. Mean-tone
(Also known as quarter-comma
meantone)

The wolf in meantone tuning is so
horrible and such an obstacle that, by the later
seventeenth century, it was being modified substantially
in practice. Modified
meantone is probably the most
appropriate temperament for most of the 'early' organ
music we now hear  even though Buxtehude and Bach
were clearly among those exploring new tuning systems,
their compositional technique remains informed by the
meantone system. Simply put, the pure thirds of meantone
are de-tuned a little bit in order to try and lessen the
wolf. Modified meantone temperament was still being used
by English organ builders, including Willis, as late as
the 1850s.(includes fifth-comma
meantone and sixth-comma meantone, the latter sometimes also known as Silbermann temperament). Naturally it allows the composer to modulate
a little more freely and frequently, perhaps permitting
an occasional excursion into five sharps or flats before
returning to a more reasonable home key.

Late in the seventeenth century
theorists started to experiment with various well-tempered systems, or circulating temperaments. The object was to finally hide the wolf,
making all keys usable. It is perfectly obvious that this
could be done by distributing the intervals equally
across the scale, but this was not the path they
took (except as an academic exercise). Why? The answer
lies in the fact that these circulating (i.e. no-wolf)
temperaments are those which allow the widest exploration
of key colour. There is every indication that
musicians of the 18th century were very happy with the
expressive possibilities offered by writing in different
keys, and sought to exploit the quite different character
of each in their writing. Temperaments of this type
include the various tunings by Werckmeister (organ expert, 1691), Kirnberger (Bach pupil, early 18thC), Neidhardt (1724) and Vallotti (c1730).
Of these systems, Werckmeister III is notable for its
purity in the best keys and its suitability for organs
with large quint mixtures (many of the fourths and fifths
are in tune); but it is irregular and bumpy in the way it
deals with modulation and key colour. Vallotti is smooth
and regular, but the key colour is generally rather mild
. In all these systems it is possible to play in any key,
though the more remote keys may sound unpleasant, and
enharmonic modulation is not always happy. Other
circulating temperaments have been devised in modern
times, almost all of them suffering from the grave defect
that they are difficult to commit to memory and therefore
difficult to use in practice (you can't tune an organ
with a book in one hand).

Finally, equal temperament.
This very obvious solution has been known since 350 BC
(!), but did not become widespread until the late 18th
century (50-100 years later in the English speaking
world). The advantages are obvious  all keys are
usable without fear or favour, and full enharmonic
modulation is possible. The disadvantages are also clear:
not one interval is dead in tune (indeed in any major
scale the thirds and leading notes are extremely sharp),
and there is no key-colour whatever. In organs, reeds
sound grittier and tierce mixtures begin to scream.

A few words on Bach
and temperament. Bach did not at any time advocate the use of
equal temperament. He wrote two sets of pieces called
'Das Wohltemperierte Klavier' ('The Well-tempered
Keyboard'), avoiding the German term for equal
temperament, which is 'Gleich-Schwebende Temperatur.'
These 48 pieces are designed to exhibit the full range of
key-colour available from a circulating temperament, and
careful examination of the texts shows that Bach varied
his compositional technique according to the key he was
writing in.

All the rest of Bach's music falls into
the more conventional patterns of the day, most of it
being quite well suited to modified mean-tone temperament
(even if it continually pushes at the boundaries of this
system). It is particularly noticeable in the organ music
that Bach uses modulation as an expressive device, and
dares to use the highly coloured remote keys for periods
of tension. Today we often wonder how 17th and 18th
century players could stand playing long pieces without
changing the stops from time to time. The answer is that
the modulation from one key to another during the course
of the music was an audible 'event', a noticeable change
 and is one of the factors that renders
registrational changes unnecessary. When Bach plunges
from a passage in the home key into a section of wild
dissonance peppered with suspensions and discords, he is deliberately
invoking the colourful effects obtainable only
from unequal temperament.

Why, then, did Bach write such works as
the Mass in B minor, in four sharps? In this instance it
is possible that the instrumentalists were playing
instruments tuned to chamber pitch, a whole tone flatter
than organ pitch. The organ continuo would be transposed
by the player, who would be playing in serene,
no-accidentals, A minor. A quick check of the score will
show that, if this transposition is assumed to have taken
place, then the continuo player will have been able to
avoid the G#  Eb wolf at all times.

Today's instrument makers and theorists
have always erred on the side of caution and have avoided
large scale use of meantone or modified meantone,
preferring to offer us more 'usable' and 'versatile'
circulating temperaments. My own feeling is that this is
a great shame, for, despite Bach's famous jokes at
Silbermann's expense (he is supposed to have played
Gottfried Silbermann's organs in outlandish keys, until
the builder was forced to retire, 'his wolf howling in
his ears'), his music is surely centered round the
meantone tradition and the many colourful possibilities
it offers.

Those who wish to hear large scale
demonstrations of major works  Bach and others
 played on old unequally tempered organs, are
recommended to try Harald Vogel's recordings of the
Schnitger at Norden and Ton Koopman's recordings of Bach,
especially on the G. Silbermann at Freiberg. Amongst
instrumental recordings I would recommend those of the
music of Monteverdi (Vespers et al.) made by Andrew Parot
and the Taverner Choir and Consort (for which I tuned the
Mander continuo organ in quarter-comma meantone).

Mention of Italian music brings us
naturally to the question of chromaticism in
early music. This is an area in which the Italians
excelled, but there are examples from many other
countries, especially in the seventeenth century, where
the chromatic scale is used for special effect. On equal
temperament this is uniformly dull, and music of this
type fails dismally to engage the listener. However, in
meantone tuning a quite different picture emerges. The
half-steps of the scale are all of differing sizes, and
the chromatic scale, far from being smooth and oily,
becomes a bumpy, eccentric and challenging affair. The
harmony it brings with it displays rapid, even
kaleidoscopic, changes of key colour through constant
modulation, and the repeated build-up and release of
harmonic tension is characteristic and fascinating.

I would also mention one other genre
especially suited to meantone  the English Trumpet
Voluntary. The acid whining of thirds and sixths in equal
temperament makes these pieces less than harmonious to
the modern ear. Even in a circulating temperament they
fail to 'come across.' In meantone they present a quite
different picture, one that I tried to hint at in my
imaginary conversation with John Stanley, posted to the
list a few months ago.

I hope that this introduction will
encourage some of you to listen out for the many positive
features of unequal temperament, even though the wide
range of the modern repertoire means that most new organs
will (and probably should) be tuned to equal temperament.

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